Victor J Katz (ed ) - The Mathematics of Egypt, Mesopotamia, China, India, and Islam_ A Sourcebook-Princeton University Press (2007) - FERNANDA NOEMI CAMPOS MENDIETA

Vista previa del material en texto

The Mathematics 
Egypt,
Mesopotamia
China,
India, and 
Islam
/
SA Sourcebook
Victor Katz, Editor
Annette Imhausen 
Eleanor Robson 
Joseph Dauben 
Kim Plofker 
J. Lennart Berggren
P R I N C E T O N U N I V E R S I T Y P R E S S * P R I N C E T O N A ND O X F O R D
Copyright © 2007 by Princeton University Press
Published by Princeton University Press, 41 William Street, 
Princeton, New Jersey 08540
In the United Kingdom: Princeton University Press, 3 Market Place, 
Woodstock, Oxfordshire 0X20 1SY
All Rights Reserved
Library of Congress Cataloging-in-Publication Data
The mathematics of Egypt, Mesopotamia, China, India, 
and Islam: a sourcebook/Victor Katz, editor, 
p. cm.
Includes bibliographical references and index.
ISBN-13: 978-0-691-11485-9 (hardcover: alk. paper)
ISBN-10: 0-691-11485-4 (hardcover: alk. paper)
1. Mathematics, Ancient-Sources. 2. Mathematics-History-Sources. 
I. Katz, Victor J.
QA22.M3735 2007
510.9-dc22 2006030851
British Library Cataloging-in-Publication Data is available
This book has been composed in Minion and Univers
Printed on acid-free paper, <x>
press.princeton.edu
Printed in the United States of America
1 3 5 7 9 10 8 6 4 2
Contents
Preface ix 
Permissions xi 
Introduction 1
C h ap te r 1
Egypt ian M a t h e ma t ic s
A n n e tte Im h au sen 
Preliminary Remarks 7
I. Introduction 9
a. Invention o f writing and number systems 13
b. Arithmetic 14
c. Metrology 17
II. Hieratic Mathematical Texts 17
a. Table texts 18
b. Problem texts 24
III. Mathematics in Administrative Texts 40
a. Middle Kingdom texts: The Reisner papyri 40
b. New Kingdom texts: Ostraca from Deir el Medina 44
IV. Mathematics in the Graeco-Roman Period 46
a. Context 46
b. Table texts 47
c. Problem texts 48
V. Appendices 52
a. Glossary of Egyptian terms 52
b. Sources 52
c. References 54
vi I C o n te n ts
Chapter 2
M e s o p o t a m i a n M a t h e ma t ic s
Eleanor Robson
I. Introduction 58
a. Mesopotamian mathematics through Western eyes 58
b. Mathematics and scribal culture in ancient Iraq 62
c. From tablet to translation 65
d. Explananda 68
II. The Long Third Millennium, c. 3200-2000 bce 73
a. Uruk in the late fourth millennium 73
b. Shuruppag in the mid-third millennium 74
c. Nippur and Girsu in the twenty-fourth century bce 76
d. Umma and Girsu in the twenty-first century bce 7 8
III. The Old Babylonian Period, c. 2000-1600 bce 82
a. Arithmetical and metrological tables 82
b. Mathematical problems 92
c. Rough work and reference lists 142
IV. Later Mesopotamia, c. 1400-150 bce 154
V. Appendices 180
a. Sources 180
b. References 181
C h a p te r 3
Chinese M a t h e ma t ic s
Jo se p h W . D auben 
Preliminary Remarks 187
I. China: The Historical and Social Context 187
II. Methods and Procedures: Counting Rods, The "Out-In"
Principle 194
III. Recent Archaeological Discoveries: The Earliest Yet-Known 
Bamboo Text 201
IV. Mathematics and Astronomy: The Zhou bi suan jing and Right 
Triangles (The Gou-gu or "Pythagorean" Theorem) 213
V. The Chinese "Euclid", Liu Hui 226
a. The Nine Chapters 227
b. The Sea Island Mathematical Classic 288
VI. The "Ten Classics" of Ancient Chinese Mathematics 29 3
a. Numbers and arithmetic: The Mathematical Classic o f Master Sun 2 95
b. The Mathematical Classic o f Zhang Qiujian 3 0 2
C o n te n ts I
VII. Outstanding Achievements of the Song and Yuan Dynasties 
(960-1368 CE) 308
a. Qin Jiushao 3 0 9
b. Li Zhi (Li Ye) 3 2 3
c. Yang Hui 3 2 9
d. Zhu Shijie .343
VIII. Matteo Ricci and Xu Guangxi, "Prefaces" to the First Chinese
Edition of Euclid's Elements (1607) 366
IX. Conclusion 375
X. Appendices 379
a. Sources 3 7 9
b. Bibliographic guides 3 7 9
c. References 3 8 0
Chapter 4
M a t he ma t ic s in India
Kim Plofker
I. Introduction: Origins of Indian Mathematics 385
II. Mathematical Texts in Ancient India 386
a. The Vedas 3 8 6
b. The Sulbasutras 3 87
c. Mathematics in other ancient texts 3 9 3
d. Number systems and numerals 3 9 5
III. Evolution of Mathematics in Medieval India 398
a. Mathematics chapters in Siddhanta texts 3 9 8
b. Transmission of mathematical ideas to the Islamic world 43 4
c. Textbooks on mathematics as a separate subject 43 5
d. The audience for mathematics education 477
e. Specialized mathematics: Astronomical and 
cosmological problems 4 7 8
IV. The Kerala School 480
a. Madhava, his work, and his school 48 0
b. Infinite series and the role of demonstrations 481
c. Other mathematical interests in the Kerala school 49 3
V. Continuity and Transition in the 
Second Millennium 4 9 8
a. The ongoing development of Sanskrit mathematics 49 8
b. Scientific exchanges at the courts of 
Delhi and Jaipur 5 0 4
c. Assimilation of ideas from Islam; mathematical table texts 5 0 6
viii I C o n te n ts
VI. Encounters with Modern Western Mathematics 507
a. Early exchanges with European mathematics 507
b. European versus “native” mathematics education in British India 5 0 8
c. Assimilation into modern global mathematics 51 0
VII. Appendices 511
a. Sources 511
b. References 512
Chapter 5
M a t h e m a t i cs in Med i e va l Islam
J. Lennart Berggren
I. Introduction 515
II. Appropriation of the Ancient Heritage 520
III. Arithmetic 525
IV. Algebra 542
V. Number Theory 560
VI. Geometry 564
a. Theoretical geometry 5 64
b. Practical geometry 61 0
VII. Trigonometry 621
VIII. Combinatorics 658
IX. On mathematics 666
X. Appendices 671
a. Sources 671
b. References 67 4
Contributors 677 
Index 681
Preface
Kim Plofker and I conceived of this book at a meeting of the Mathematical Association of 
America several years ago. We agreed that, since there was now available a fairly large collec­
tion of English translations of mathematical texts from Egypt, Mesopotamia, China, India, 
and Islam, the time was ripe to put together an English sourcebook. This book would have 
texts gathered together so that they could easily be studied by all those interested in the math­
ematics of ancient and medieval times. I secured commitments to edit the five sections from 
outstanding scholars with whom I was familiar, scholars who had already made significant 
contributions in their fields, and were fluent in both the original languages of the texts and 
English. There are a growing number of scholars investigating the works of these civilizations, 
but I believe that this group was the appropriate one to bring this project to fruition.
The editors decided that to the extent possible we would use already existing English trans­
lations with the consent of the original publisher, but would, where necessary, produce new 
ones. In certain cases, we decided that an existing translation from the original language into 
French or German could be retranslated into English, but the retranslation was always made 
with reference to the original language. As it turned out, the section editors of both the 
Egyptian and Mesopotamian sections decided to produce virtually all new translations, 
because they felt that many previous translations had been somewhat inadequate. The other 
sections have a mix of original translations and previously translated material.
Each of the five sections of the Sourcebook has a preface, written by the section editor, giv­
ing an overview of the sources as well as detailing the historical setting of the mathematics in 
that civilization. The individual sources themselves also have introductions. In addition, many 
of the sources contain explanations to help the reader understand the sometimes fairly cryptic 
texts. In particular, the Chinese and Indian sections have considerably more detailed explana­
tions than the Islamic section, because Islamic mathematicians, being well schooled in Greek 
mathematics, use “our” techniques of mathematical analysis and proof. Mathematicians from 
China and India, and from Egypt and Mesopotamia as well, come from traditions far different 
from ours. So the editors of those sections have spent considerable effortin guiding the reader 
through this “different” mathematics.
The book is aimed at those having knowledge of mathematics at least equivalent to a U.S. 
mathematics major. Thus, the intended audience of the book includes students studying
x I P re face
mathematics and the history of mathematics, mathematics teachers at all levels, mathemati­
cians, and historians o f mathematics.
The editors of this Sourcebook wish to thank our editors at Princeton University Press, 
David Ireland and Vickie Kearn, who have been extremely supportive in guiding this book 
from concept to production. We also wish to thank Dale Cotton, our production editor, 
Dimitri Karetnikov, the illustration specialist, and Alison Anderson, the copy editor, for their 
friendly and efficient handling of a difficult manuscript. Finally, as always, I want to thank my 
wife Phyllis for her encouragement and for everything else.
Victor J. Katz 
Silver Spring, MD 
December, 2005
Permissions
Chapter 1
section iv: The material from the Demotic papyri is reprinted from Richard Parker, Demotic 
Mathematical Papyri © 1972 by Brown University Press, by permission of University Press of 
New England, Hanover, NH.
Chapter 3
section iv: The material from the Zhou hi suan jing is reprinted from Astronomy and 
Mathematics in Ancient China: The Zhou Bi Suan Jing, by Christopher Cullen, © 1996, with the 
permission of Cambridge University Press.
section v: The material from the Nine Chapters on the Mathematical Art and the Sea Island 
Mathematical Manual is reprinted from The Nine Chapters on the Mathematical Art: 
Companion and Commentary, by Shen Kangshen, John Crossley, and Anthony W.-C. Lun, 
© 1999, with the permission of Oxford University Press.
section vi.a.: The material from the Sunzi suan jing is reprinted from Fleeting Footsteps: 
Tracing the Conception o f Arithmetic and Algebra in Ancient China, by Lam Lay Yong and Ang 
Tian Se, © 1992, with the permission of World Scientific Publishing Co. Pte. Ltd.
section vi.b .: The material from the Zhang qiujian suan jing is reprinted from “Zhang 
Qiujian Suanjing (The Mathematical Classic of Zhang Qiujian): An Overview,” by Lam Lay 
Yong, Archive for History o f Exact Sciences 50 (1997), 201-240, with the kind permission of 
Springer Science and Business Media.
section vii.a: The material from chapters 1, 2, 4 of the Shu shu jiu zhang is reprinted from 
Chinese Mathematics in the Thirteenth Century: The Shu-shu chiu-chang o f Ch’in Chiu-shao, by 
Ulrich Libbrecht, © 1973, with the permission of MIT Press. The material from chapter 3 of 
the Shu shu jiu zhang is reprinted from A History o f Chinese Mathematics, by Jean-Claude 
Martzloff, Stephen Wilson, trans., © 1997, with the kind permission of Springer Science and 
Business Media.
xii I P erm iss io ns
section vii.b: The material from Li Zhi’s Ce yuan hai jing is reprinted from “Chinese 
Polynomial Equations in the Thirteenth Century,” in Li Guihao et a l, eds., Explorations in the 
History o f Science and Technology in China, Chinese Classics Publishing House, Shanghai, 
© 1982. The material from Li Zhi’s Yigu yanduan is reprinted from “Li Ye and his Yi Gu Yan 
Duan (Old Mathematics in Expanded Sections),” by Lam Lay Yong and Ang Tian Se, Archive 
for History o f Exact Sciences 29 (1984), 237-265 with the kind permission of Springer Science 
and Business Media.
section vii.c.: The material from Yang Hui’s Xiangjie Jiuzhang Suanfa is reprinted from “On 
The Existing Fragments of Yang Hui’s Hsiang Chieh Suan Fa,” by Lam Lay Yong, Archive for 
History o f Exact Sciences 6 (1969), 82-88 with the kind permission of Springer Science and 
Business Media. The material from Yang Hui’s Yang Hui Suan Fa is reprinted from A Critical 
Study o f the Yang Hui Suan Fa, by Lam Lay Yong, © 1977, with the permission of Singapore 
University Press.
section vm : The material from Matteo Ricci and Xu Guangxi’s, “Prefaces” to the First 
Chinese Edition of Euclid’s Elements is reprinted from Euclid in China, by Peter M. Engelfriet, 
© 1998 by permission of Brill Academic Publishers.
Chapter 4
section ii.b: The material from the Sulbasutras is reprinted from The Sulbasutras, by 
S. N. Sen and A. K. Bag, © 1983 by permission of the Indian National Science Academy.
section ii.d.: Figure 2.5 is reprinted from The Concept ofSunya, by A. K. Bag and S. R. Sarma, 
eds., © 2003 by permission of the Indian National Science Academy.
section iii.a.: The material from the Aryabhatiya is reprinted from Expounding the 
Mathematical Seed: a Translation ofBhaskara I on the Mathematical Chapter o f the Aryabhatiya, 
by Agathe Keller, © 2005 by permission of Birkhauser Verlag AG.
section iii.c.: The material from the BakhshatiManuscript is reprinted from The Bakhshali 
Manuscript: An Ancient Indian Mathematical Treatise, © 1995 by permission of Egbert 
Forsten Publishing. The material from the Byaganita of Bhaskara II with commentary of 
Suryadasa is reprinted from The Suryaprakdsa o f Suryadasa: A Commentary on Bhaskaracarya’s 
BTjaganita, by Pushpa Kumari fain, © 2001 by permission of the Oriental Institute in 
Vadodara.
section iii.e.: The material from the Sisyadhivrddhidatantra of Lalla is reprinted from 
Sisyadhivrddhida Tantra o f Lalla, by Bina Chatterjee, © 1981 by permission of the Indian 
National Science Academy.
section iv .b .: The material from the Kriyakramakari of Sankara is reprinted from Studies in 
Indian Mathematics: Series, Pi and Trigonometry, by T. Hayashi, T. Kusuba, and M. Yano, 
Setsuro Ikeyama, trans., by permission of the authors.
section v.a.: The material from the Ganitakaumudioi Narayana Pandita is reprinted from 
Combinatorics and Magic Squares in India: A Study o f Narayana Pandita’s “Ganitakaumudi) 
chapters 13-14, a Brown University dissertation by Takanori Kusuba and is used by permission 
of the author.
P erm iss io n s I xiii
Chapter 5
section n: The material from the Banu Musa On Conics is reprinted from Apollonius’ Conics 
Books V-VII: The Arabic Translation o f the Lost Greek Original in the Version o f the Banu Musa, 
by G. I. Toomer, © 1990 with the kind permission of Springer Science and Business Media. The 
material from ibn al-Haytham’s Completion o f the Conics is reprinted from Ibn al-Haythams 
Completion o f the Conics, by J. P. Hogendijk, © 1985 with the kind permission of Springer 
Science and Business Media.
section hi: The material from al-Khwarizml’s Treatise on Hindu Calculation is reprinted from 
“Thus Spake al-Khwarizml: A Translation of the Text of Cambridge University Library Ms. 
Ik.vi.5,” by lohn Crossley and Alan Henry, Historia Mathematica 17 (1990), 103-131 with per­
mission from Elsevier Ltd. The material from al-Uqlldisl’s Chapters on Hindu Arithmetic is 
reprinted from The Arithmetic o f al-Uqlidisi, by A. Saidan, © 1978 by permission of Reidel 
Publishers. The material from Kushyar ibn Laban’s Principles o f Hindu Reckoning is reprinted 
from Kushyar ibn Laban Principles o f Hindu Reckoning, by M. Levey and M. Petruck, © 1966 
by permission of The University of Wisconsin Press.
section iv: The material from al-Samaw’al on al-Karajl and binomial coefficients is reprinted 
from The Development o f Arabic Mathematics: Between Arithmetic and Algebra, by Roshdi 
Rashed, © 1994 with the kind permission of Springer Science and Business Media. The mate­
rial from the algebra of Omar Khayyam is reprinted from The Algebra o f Omar Khayyam, by 
D. Kasir, © 1931 with the permission of Columbia Teachers College Press.
section v: The material from al-Baghdadl on balanced numbers is reprinted from “Two 
Problems of Number Theory in Islamic Times,” by J. Sesiano, Archive for History o f Exact 
Sciences 41, (1991), 235-238 with the kind permission of Springer Science and Business Media.
section vi.a.: The material from al-Kuhl and al-Sabl on centers of gravity is reprinted from 
“The Correspondence of Abu Sahl al-Kuhl and Abu Ishaq al-Sabl,” by I. Lennart Berggren, 
lournalfor the Historyo f Arabic Science 7, (1983), 39-124 with the permission of the Journal. 
The material from al-Kuhl on the astrolabe is reprinted from “Abu Sahl al-Kuhl’s Treatise 
on the Construction of the Astrolabe with Proof: Text, Translation and Commentary,” by 
J. Lennart Berggren, Physis 31 (1994), 141-252 with the permission o f Leo S. Olschki, pub­
lisher. The material from al-SijzT on the construction of the heptagon is reprinted from 
“Greek and Arabic Constructions of the Regular Heptagon,” by ]. P. Hogendijk, Archive for 
History o f Exact Sciences 30 (1984), 197-330 with the kind permission of Springer Science 
and Business Media. The material on the anonymous treatise on the nine-sided figure is 
reprinted from “An Anonymous Treatise on the Regular Nonagon,” by I. L. Berggren, Journal 
for the History o f Arabic Science 5 (1981),37-41 with the permission of the Journal The mate­
rial from al-Jazarl on constructing geometrical instruments is reprinted from The Book 
o f Knowledge o f Ingenious Mechanical Devices by Ibn al-Razzaz al-Jazan, by Donald Hill, 
© 1974 by permission of Reidel Publishers. The material from al-TusI on the theory of par­
allels is reprinted from A History o f Non-Euclidean Geometry: Evolution o f the Concept o f a 
Geometric Space, by B. A. Rosenfeld, © 1988 with the kind permission of Springer Science 
and Business Media.
section vi.b .: The material from Abu al-Wafa’ on the geometry of artisans is reprinted from 
“Mathematics and Arts: Connections between Theory and Practice in the Medieval Islamic
x iv I P erm iss io ns
World”, by Alpay Ozdural, Historia Mathematica 27, (2000), 171-201 with permission from 
Elsevier, Ltd. The two patterns for muqarnas are reprinted from Topkapi Scroll-Geometry and 
Ornament in Islamic Architecture, by G. Necipoglu, © 1995 with permission of the Getty 
Research Institute.
section vn: The material from al-BirunTon geodesy is reprinted from The Determination o f 
the Coordinates o f Cities: Al-BirunT’s Tahdtd al-Amakin, by Jamil Ali, © 1967 with permission 
of the American University of Beirut Press, Beirut, Lebanon. The material from al-Kashl on 
interpolation using second order differences is reprinted from “A Medieval Interpolation 
Scheme Using Second Order Differences,” by E. S. Kennedy, in Studies in the Islamic Exact 
Sciences by E. S. Kennedy, Colleagues, and Former Students, edited by D. A. King and M. H. 
Kennedy, © 1983 with permission of the American University of Beirut Press, Beirut, Lebanon.
section vm : The material from al-BirunTon the number of possibilities for eclipse data is 
reprinted from The Determination o f the Coordinates o f Cities: Al-BirunT’s TahdTd al-Amakin, 
by Jamil Ali, © 1967 with permission of the American University of Beirut Press, Beirut, 
Lebanon.
section ix: The material from al-Kuhlon the certainty deriving from mathematics is reprinted 
from “The Correspondence of Abu Sahl al-Kuhl and Abu Ishaq al-Sabl: A Translation with 
commentaries,” by J. Lennart Berggren, Journal for the History o f Arabic Science 7, (1983), 
39-124 with the permission of the Journal. The material from al-SijzT on heuristics is reprinted 
from Al-SijzT’s Treatise on Geometrical Problem Solving, by J. P. Hogendijk, with the permission 
of the author.
The Mathematics of
Egypt,
Mesopotamia,
China,
India, and 
Islam
JA Sourcebook
i
Introduction
A century ago, mathematics history began with the Greeks, then skipped a thousand years and 
continued with developments in the European Renaissance. There was sometimes a brief 
mention that the “Arabs” preserved Greek knowledge during the dark ages so that it was avail­
able for translation into Latin beginning in the twelfth century, and perhaps even a note that 
algebra was initially developed in the lands of Islam before being transmitted to Europe. 
Indian and Chinese mathematics barely rated a footnote.
Since that time, however, we have learned much. First o f all, it turned out that the Greeks 
had predecessors. There was mathematics both in ancient Egypt and in ancient 
Mesopotamia. Archaeologists discovered original material from these civilizations and deci­
phered the ancient texts. In addition, the mathematical ideas stemming from China and 
India gradually came to the attention of historians. In the nineteenth century, there had been 
occasional mention o f these ideas in fairly obscure sources in the West, and there had even 
been translations into English or other western languages of certain mathematical texts. But 
it was only in the late twentieth century that major attempts began to be made to understand 
the mathematical ideas of these two great civilizations and to try to integrate them into a 
worldwide history of mathematics. Similarly, the nineteenth century saw numerous transla­
tions of Islamic mathematical sources from the Arabic, primarily into French and German. 
But it was only in the last half of the twentieth century that historians began to put together 
these mathematical ideas and attempted to develop an accurate history of the mathematics 
of Islam, a history beyond the long-known preservation of Greek texts and the algebra of al- 
Khwarizmi. Yet, even as late as 1972, Morris Kline’s monumental work Mathematical Thought 
from Ancient to Modern Times contained but 12 pages on Mesopotamia, 9 pages on Egypt, 
and 17 pages combined on India and the Islamic world (with nothing at all on China) in its 
total of 1211 pages.
It will be useful, then, to give a brief review of the study of the mathematics of Egypt, 
Mesopotamia, China, India, and Islam to help put this Sourcebook in context.
To begin with, our most important source on Egyptian mathematics, the Rhind 
Mathematical Papyrus, was discovered, probably in the ruins of a building in Thebes, in the 
middle of the nineteenth century and bought in Luxor by Alexander Henry Rhind in 1856. 
Rhind died in 1863 and his executor sold the papyrus, in two pieces, to the British Museum 
in 1865. Meanwhile, some fragments from the break turned up in New York, having been 
acquired also in Luxor by the American dealer Edwin Smith in 1862. These are now in the
2 I V ic to r Katz
Brooklyn Museum. The first translation of the Rhind Papyrus was into German in 1877. The 
first English translation, with commentary, was made in 1923 by Thomas Peet of the 
University of Liverpool. Similarly, the Moscow Mathematical Papyrus was purchased around 
1893 by V. S. Golenishchev and acquired about twenty years later by the Moscow Museum of 
Fine Arts. The first notice of its contents appeared in a brief discussion by B. A. Turaev, con­
servator of the Egyptian section of the museum, in 1917. He wrote chiefly about problem 14, 
the determination of the volume of a frustum of a square pyramid, noting that this showed 
“the presence in Egyptian mathematics of a problem that is not to be found in Euclid.” The 
first complete edition of the papyrus was published in 1930 in German by W. W. Struve. 
The first complete English translation was published by Marshall Clagett in 1999.
Thus, by early in the twentieth century, the basic outlines of Egyptian mathematics were 
well understood— at least the outlines as they could be inferred from these two papyri. And 
gradually the knowledge of Egyptian mathematics embedded in these papyri and other 
sources became part of the global story of mathematics, with one of the earliest discussions 
being in Otto Neugebauer’s Vorlesungen tiber Geschichte der antiken Mathematischen 
Wissenschaften (more usually known as Vorgriechische Mathematik) of 1934, and further 
discussions and analysis by B. L. Van der Waerden in his Science Awakening o f 1954. A more 
recent survey is by James Ritter in Mathematics Across Cultures.
A similar story can be told about Mesopotamian mathematics. Archaeologists had begun 
to unearth the clay tablets of Mesopotamia beginning in the middle of the nineteenth century, 
and it was soon realized that some of the tablets contained mathematicaltables or problems. 
But it was not until 1906 that Hermann Hilprecht, director of the University o f Pennsylvania’s 
excavations in what is now Iraq, published a book discussing tablets containing multiplication 
and reciprocal tables and reviewed the additional sources that had been published earlier, if 
without much understanding. In 1907, David Eugene Smith brought some of Hilprecht’s work 
to the attention of the mathematical world in an article in the Bulletin o f the American 
Mathematical Society, and then incorporated some of these ideas into his 1923 History o f 
Mathematics.
Meanwhile, other archaeologists were adding to Hilprecht’s work and began publishing 
some of the Mesopotamian mathematical problems. The study of cuneiform mathematics 
changed dramatically, however, in the late 1920s, when Francois Thureau-Dangin and Otto 
Neugebauer independently began systematic programs of deciphering and publishing these 
tablets. In particular, Neugebauer published two large collections: Mathematische Keilschrift- 
Texte in 1935-37 and (with Abraham Sachs) Mathematical Cuneiform Texts in 1945. He then 
summarized his work for the more general mathematical public in his 1951 classic, The Exact 
Sciences in Antiquity. Van der Waerden’s Science Awakening was also influential in publicizing 
Mesopotamian mathematics. Jens Hoyrup’s survey of the historiography of Mesopotamian 
mathematics provides further details.
Virtually the first mention of Chinese mathematics in a European language was in several 
articles in 1852 by Alexander Wylie entitled “lottings on the Science of the Chinese: 
Arithmetic,” appearing in the North China Herald, a rather obscure Shanghai journal. 
However, they were translated in part into German by Karl L. Biernatzki and published in 
Crelles Journal in 1856. Six years later they also appeared in French. It was through these arti­
cles that Westerners learned of what is now called the Chinese Remainder problem and how 
it was initially solved in fourth-century China, as well as about the ten Chinese classics and 
the Chinese algebra of the thirteenth century. Thus, by the end of the nineteenth century,
In tro d u c tio n 1 3
European historians of mathematics could write about Chinese mathematics, although, since 
they did not have access to the original material, their works often contained errors.
The first detailed study of Chinese mathematics written in English by a scholar who could 
read Chinese was Mathematics in China and Japan, published in 1913 by the Japanese scholar 
Yoshio Mikami. Thus David Eugene Smith, who co-authored a work solely on Japanese math­
ematics with Mikami, could include substantial sections on Chinese mathematics in his own 
History of 1923. Although other historians contributed some material on China during the 
first half of the twentieth century, it was not until 1959 that a significant new historical study 
appeared, volume 3 of Joseph Needham’s Science and Civilization in China, entitled 
Mathematics and the Sciences o f the Heavens and the Earth. One of Needham’s chief collabora­
tors on this work was Wang Ling, a Chinese researcher who had written a dissertation on the 
Nine Chapters at Cambridge University. Needham’s work was followed by the section on 
China in A. R Yushkevich’s history of medieval mathematics (1961) in Russian, a book that 
was in turn translated into German in 1964. Since that time, there has been a concerted effort 
by both Chinese and Western historian of mathematics to make available translations of the 
major Chinese texts into Western languages.
The knowledge in the West of Indian mathematics occurred much earlier than that of 
Chinese mathematics, in part because the British ruled much of India from the eighteenth cen­
tury on. For example, Henry Thomas Colebrooke collected Sanskrit mathematical and astro­
nomical texts in the early nineteenth century and published, in 1817, his Algebra with 
Arithmetic and Mensuration from the Sanscrit o f Brahmegupta and Bhascara. Thus parts of the 
major texts of two of the most important medieval Indian mathematicians were available in 
English, along with excerpts from Sanskrit commentaries on these works. Then in 1835, 
Charles Whish published a paper dealing with the fifteenth-century work in Kerala on infinite 
series, and Ebenezer Burgess in 1860 published a translation of the Surya-siddhanta, a major 
early Indian work on mathematical astronomy. Hendrik Kern in 1874 produced an edition of 
the Aryabhatiya of Aryabhata, while George Thibaut wrote a detailed essay on the Sulbasutras, 
which was published, along with his translation of the Baudhayana Sulbasutra, in the late 1870s. 
The research on medieval Indian mathematics by Indian researchers around the same time, 
including Bapu Deva Sastrl, Sudhakara DvivedI, and S. B. Dikshit, although originally published 
in Sanskrit or Hindi, paved the way for additional translations into English.
Despite the availability of some Sanskrit mathematical texts in English, it still took many 
years before Indian contributions to the world of mathematics were recognized in major 
European historical works. Of course, European scholars knew about the Indian origins of the 
decimal place-value system. But in part because of a tendency in Europe to attribute Indian 
mathematical ideas to the Greeks and also because of the sometimes exaggerated claims by 
Indian historians about Indian accomplishments, a balanced treatment of the history of 
mathematics in India was difficult to achieve. Probably the best of such works was the History 
o f Indian Mathematics: A Source Book, published in two volumes by the Indian mathemati­
cians Bibhutibhusan Datta and Avadhesh Narayan Singh in 1935 and 1938. In recent years, 
numerous Indian scholars have produced new Sanskrit editions of ancient texts, some of 
which have never before been published. And new translations, generally into English, are also 
being produced regularly, both in India and elsewhere.
As to the mathematics of Islam, from the time of the Renaissance Europeans were aware 
that algebra was not only an Arabic word, but also essentially an Islamic creation. Most early 
algebra works in Europe in fact recognized that the first algebra works in that continent were
4 I V ic to r Katz
translations of the work of al-Khwarizml and other Islamic authors. There was also some 
awareness that much of plane and spherical trigonometry could be attributed to Islamic 
authors. Thus, although the first pure trigonometrical work in Europe, On Triangles by 
Regiomontanus, written around 1463, did not cite Islamic sources, Gerolamo Cardano noted 
a century later that much of the material there on spherical trigonometry was taken from the 
twelfth-century work of the Spanish Islamic scholar Jabir ibn Aflah.
By the seventeenth century, European mathematics had in many areas reached, and in 
some areas surpassed, the level of its Greek and Arabic sources. Nevertheless, given the con­
tinuous contact of Europe with Islamic countries, a steady stream of Arabic manuscripts, 
including mathematical ones, began to arrive in Europe. Leading universities appointed pro­
fessors of Arabic, and among the sources they read were mathematical works. For example, the 
work of Sadr al-TusI (the son of NasTr al-Din al-TusI) on the parallel postulate, written origi­
nally in 1298, was published in Rome in 1594 with a Latin title page. This work was studied 
by lohn Wallis in England, who then wrote about its ideas as he developed his own thoughts 
on the postulate. Still later, Newton’s friend, Edmond Halley, translated into Latin 
Apollonius’s Cutting-off o f a Ratio, a work that had been lost in Greek but had been preserved 
via an Arabic translation.
Yet in the seventeenth and eighteenth centuries, when Islamic contributions to mathemat­
ics may well have helped Europeans develop their own mathematics, most Arabic manuscripts 
lay unread in libraries around the world. It was not until the mid-nineteenthcentury that 
European scholars began an extensive program of translating these mathematical manu­
scripts. Among those who produced a large number of translations, the names of Heinrich 
Suter in Switzerland and Franz Woepcke in France stand out. (Their works have recently been 
collected and republished by the Institut fur Geschichte der arabisch-islamischen 
Wissenschaften.) In the twentieth century, Soviet historians of mathematics began a major 
program of translations from the Arabic as well. Until the middle of the twentieth century, 
however, no one in the West had pulled together these translations to try to give a fuller pic­
ture of Islamic mathematics. Probably the first serious history of Islamic mathematics was a 
section of the general history of medieval mathematics written in 1961 by A. P. Yushkevich, 
already mentioned earlier. This section was translated into French in 1976 and published as a 
separate work, Les mathematiques arabes ( VIIIe-XVe siecles). Meanwhile, the translation pro­
gram continues, and many new works are translated each year from the Arabic, mostly into 
English or French.
By the end of the twentieth century, all o f these scholarly studies and translations of the 
mathematics of these various civilizations had an impact on the general history of mathe­
matics. Virtually all recent general history textbooks contain significant sections on the math­
ematics of these five civilizations. As this sourcebook demonstrates, there are many ideas that 
were developed in these five civilizations that later reappeared elsewhere. The question that 
then arises is how much effect the mathematics of these civilizations had on what is now world 
mathematics of the twenty first-century. The answer to this question is very much under 
debate. We know of many confirmed instances of transmission of mathematical ideas from 
one of these cultures to Europe or from one of these cultures to another, but there are numer­
ous instances where, although there is circumstantial evidence of transmission, there is no 
definitive documentary evidence. Whether such will be found as more translations are made 
and more documents are uncovered in libraries and other institutions around the world is a 
question for the future to answer.
In tro d u c tio n I 5
References
Biernatzki, Karl L. 1856. “Die Arithmetic der Chinesen.” Crelles Journal 52: 59-94.
Burgess, Ebenezer. 1860. “The Suryasiddhanta.” Journal o f the American Oriental Society, 6.
Clagett, Marshall. 1999. Ancient Egyptian Science: A Source Book. Vol. 3, Ancient Egyptian 
Mathematics. Philadelphia: American Philosophical Society.
Colebrooke, Henry Thomas. 1817. Algebra with Arithmetic and Mensuration from the Sanscrit 
o f Brahmegupta and Bhascara. London: John Murray.
Datta, B., and A. N. Singh., 1935/38. History o f Hindu Mathematics: A Source Book. 2 vols. 
Bombay: Asia Publishing House.
Hilprecht, Hermann V. 1906. Mathematical, Metrological and Chronological Tablets from the 
Temple Library o f Nippur Philadelphia: University Museum o f Pennsylvania.
Hoyrup, Jens. 1996. “Changing Trends in the Historiography of Mesopotamian Mathematics: 
An Insider’s View.” History o f Science 34: 1-32.
Kern, Hendrik. 1874. The Aryabhatiya, with the Commentary Bhatadipika o f Paramadicyara. 
Leiden: Brill.
Kline, Morris. 1972. Mathematical Thought from Ancient to Modern Times. New York: Oxford 
University Press.
Mikami, Yoshio. 1913. The Development o f Mathematics in China and Japan. Leipzig: Teubner.
Needham, Joseph. 1959. Science and Civilization in China. Vol. 3, Mathematics and the Sciences 
o f the Heavens and the Earth. Cambridge: Cambridge University Press.
Neugebauer, Otto. 1934. Vorlesungen liber Geschichte der antiken Mathematischen 
Wissenschaften. I. Vorgriechische Mathematik. Berlin: Springer-Verlag.
---------- . 1935-37. Mathematische Keilschrift-Texte. Berlin: Springer-Verlag.
---------- . 1945. Mathematical Cuneiform Texts (with Abraham Sachs). New Haven: American
Oriental Society.
---------- . 1951. The Exact Sciences in Antiquity. Princeton: Princeton University Press.
Peet, Thomas. 1923. The Rhind Mathematical Papyrus, British Museum 10057 and 10058. 
Introduction, Transcription, Translation and Commentary. London.
Ritter, James. 2000. “Egyptian Mathematics.” In Helaine Selin, ed., Mathematics Across 
Cultures: The History o f Non-Western Mathematics. Dordrecht: Kluwer.
Smith, David Eugene. 1907. “The Mathematical Tablets of Nippur.” Bulletin o f the American 
Mathematical Society 13: 392-398.
-----------. 1923. History o f Mathematics. 2 vols. Boston: Ginn.
Struve, W. W. 1930. Mathematische Papyrus des Staatlichen Museums der Schonen Kiinste in 
Moskau. Quellen und Studien zur Geschichte der Mathematik A l. Berlin: Springer-Verlag.
Suter, Heinrich. 1986. Beitrage zur Geschichte der Mathematik und Astronomie im Islam. 2 vols. 
Frankfurt: Institut fur Geschichte der arabisch-islamischen Wissenschaften.
Thibaut, George. 1875. “On the Sulba-sutra.” Journal o f the Asiatic Society o f Bengal. 44: 227-275.
6 I V ic to r K atz
Thureau-Dangin, Francois. 1983. Textes mathematiques babyloniens (Ex Oriente Lux 1). 
Leiden: Brill.
Turaev, Boris. A. 1917. “The Volume of the Truncated Pyramid in Egyptian Mathematics.” 
Ancient Egypt 3 :1 0 0-102 .
Van der Waerden, B. L. 1954. Science Awakening. New York: Oxford University Press.
Whish, Charles. 1835. “On the Hindu quadrature o f a circle.” Transactions o f the Royal Asiatic 
Society o f Great Britain and Ireland 3: 509-23.
Woepcke, Franz. 1986. Etudes sur les mathematiques arabo-islamiques. 2 vols. Frankfurt: 
Institut fur Geschichte der arabisch-islamischen Wissenschaften.
Wylie, Alexander. 1852. “Jottings on the Science o f the Chinese Arithmetic.” North China 
Herald, Aug.-Nov. nos. 108-13, 116-17, 119-21.
Yushkevich, A. P. 1961. The History o f Mathematics in the Middle Ages (in Russian). Moscow: 
Fizmatgiz.
-----------. 1964. Geschichte der Mathematik in Mittelalter, Leipzig: Teubner.
-----------. 1976. Les mathematiques arabes (VIIIe -XVs siecles). Paris: Vrin.
1 Egyptian Mathematics
A n n e tte Im h a u s e n
Preliminary Remarks
The study of Egyptian mathematics is as fascinating as it can be frustrating. The preserved 
sources are enough to give us glimpses of a mathematical system that is both similar to some 
of our school mathematics, and yet in some respects completely different. It is partly this sim­
ilarity that caused early scholars to interpret Egyptian mathematical texts as a lower level of 
Western mathematics and, subsequently, to “translate” or rather transform the ancient text 
into a modern equivalent. This approach has now been widely recognized as unhistorical and 
mostly an obstacle to deeper insights. Current research attempts to follow a path that is 
sounder historically and methodologically. Furthermore, writers of new works can rely on 
progress that has been made in Egyptology (helping us understand the language and context 
of our texts) as well as in the history of mathematics.
However, learning about Egyptian mathematics will never be an easy task. The main 
obstacle is the shortage of sources. It has been over 70 years since a substantial new Egyptian 
mathematical text was discovered. Consequently, we must be extremely careful with our 
general evaluation of Egyptian mathematics. If we arbitrarily chose six mathematical publi­
cations of the past 300 years, what would we be able to say about mathematical achievements 
between 1700 and 2000 ce? This is exactly our situation for the mathematical texts of the 
Middle Kingdom (2119-1794/93 bce). On the positive side, it must be said that the available 
source material is as yet far from being exhaustively studied, and significant and fascinating 
new insights are still likely to be gained. Also, the integration of other texts that contain 
mathematical information helps to fill out the picture. The understanding of Egyptian math­
ematics dependson our knowledge of the social and cultural context in which it was created, 
used, and developed. In recent years, the use of other source material, which contains direct 
or indirect information about Egyptian mathematics, has helped us better understand the 
extant mathematical texts.
This chapter presents a selection of sources and introduces the characteristic features of 
Egyptian mathematics. The selection is taken from over 3000 years of history. Consequently, 
the individual examples have to be taken within their specific context. The introduction fol­
lowing begins with a text about mathematics from the New Kingdom (1550-1070/69 b c e ) to 
illustrate the general context of mathematics within Egyptian culture.
8 I A n n e tte Im h au sen
To introduce this text, we need to bear in mind that the development and use of mathe­
matical techniques began around 1500 years earlier with the invention of writing and number 
systems. The available evidence points to administrative needs as the motivation for this 
development. Quantification and recording of goods also necessitated the development of 
metrological systems, which can be attested as early as the Old Kingdom and possibly earlier. 
Metrological systems and mathematical techniques were used and developed by the scribes, 
that is, the officials working in the administration of Egypt. Scribes were crucial to ensuring 
the smooth collection and distribution of available goods, thus providing the material basis 
for a prospering government under the pharaoh. Evidence for mathematical techniques comes 
from the education and daily work life of these scribes. The most detailed information can be 
gained from the so-called mathematical texts, papyri that were used in the education of jun­
ior scribes. These papyri contain collections of problems and their solutions to prepare the 
scribes for situations they were likely to face in their later work.
The mathematical texts inform us first of all about different types of mathematical prob­
lems. Several groups of problems can be distinguished according to their subject. The majority 
are concerned with topics from an administrative background. Most scribes were probably 
occupied with tasks of this kind. This conclusion is supported by illustrations found on the 
walls of private tombs. Very often, in tombs of high officials, the tomb owner is shown as an 
inspector in scenes of accounting of cattle or produce, and sometimes several scribes are 
depicted working together as a group. It is in this practical context that mathematics was devel­
oped and practised. Further evidence can be found in three-dimensional models representing 
scenes of daily life, which regularly include the figure of one or more scribes. Several models 
depict the filling of granaries, and a scribe is always present to record the respective quantities.
While mathematical papyri are extant from two separate periods only, depictions of scribes 
as accountants (and therefore using mathematics) are evident from all periods beginning with 
the Old Kingdom. Additional evidence for the same type of context for mathematics appears 
during the New Kingdom in the form of literary texts about the scribal profession. These texts 
include comparisons of a scribe’s duties to duties of other professions (soldier, cobbler, farmer, 
etc.). It is clear that many of the scribes’ duties involve mathematical knowledge. The intro­
duction begins with a prominent example from this genre.
Another (and possibly the only other) area in which mathematics played an important role 
was architecture. Numerous extant remains of buildings demonstrate a level of design and 
construction that could only have been achieved with the use of mathematics. However, which 
instruments and techniques were used is not known nor always easy to discern. Past‘histori­
ography has tended to impose modern concepts on the available material, and it is only 
recently that a serious reassessment of this subject has been published.1 Again, detailed 
accounts of mathematical techniques related to architecture are only extant from the Middle 
Kingdom on. However, a few sketches from the Old Kingdom have survived as well, which 
indicate that certain mathematical concepts were present or being developed. These concepts 
then appeared fully formed in the mathematical texts.
Throughout this chapter Egyptian words appear in what Egyptologists call “transcription.” 
The Egyptian script noted only consonants (although we pronounce some of them as vowels 
today). For this reason, transcribing hieratic or hieroglyphic texts means to transform the
See [Rossi 2004].
Egyptian M a th e m a tic s I 9
text into letters which are mostly taken from our alphabet and seven additional letters 
Q, r, A, h, h, t, d ). In order to be able to read Egyptian, Egyptologists therefore agreed on the 
convention to insert (in speaking) short “e” sounds where necessary. The pronunciation of the 
Egyptian transcription alphabet is given below. This is a purely modern convention— how 
Egyptian was pronounced originally is not known. The Appendix contains a glossary of 
all Egyptian words in this chapter and their (modern) pronunciation.
Letter Pronunciation
?) a
j i or j
r a
w u, u, or w
b b
P P
f f
m m
n n
r r
h h
h like Arabic h
b like ch in “loch”
h like German ch in “ich”
z voiced s
s unvoiced s
s sh
q emphatic k
k k
8 g
t t
t like ch in “touch”
d d
d dg as in “judge“
I. Introduction
The passage below is taken from Papyrus Anastasi I,2 an Egyptian literary text of the New 
Kingdom (1550-1070/69 b c e ). This composition is a fictional letter, which forms part of a 
debate between two scribes. The letter begins, as is customary for Egyptian letters, with the 
writer Hori introducing himself and then addressing the scribe Amenemope (by the shortened 
form Mapu). After listing the necessary epithets and wishing the addressee well, Hori recounts 
receiving a letter of Amenemope, which Hori describes as confused and insulting. He then
2The hieroglyphic transcription of the various extant sources can be found in [Fischer-Elfert 1983]. An English
translation of the complete text is [Gardiner 1911]; this however is based on only ten of the extant 80 sources. The edi-
tio princeps is [Fischer-Elfert 1986]. The translation given here is my own, which is based on the work by Fischer-Elfert.
10 I A n n e tte Im h au sen
proposes a scholarly competition covering various aspects of scribal knowledge. The letter ends 
with Hori criticizing the letter of his colleague again and suggesting to him that he should sit 
down and think about the questions of the competition before trying to answer them.
The mathematical section of the letter, translated below, comprises several problems similar to 
the collections of problems found in mathematical papyri. However, in this letter, the problems 
are framed by Hori’s comments (and sometimes insults), addressed to his colleague Amenemope 
(Mapu). Hori points out several times the official position which Amenemope claims for himself 
(“commanding scribe of the soldiers,” “royal scribe”) and teases him by calling him ironically 
“vigilant scribe,” “scribe keen of wit,” “sapient scribe,” directly followed by a description of 
Amenemope’s ineptness. In between, Hori describes several situations in which Amenemope 
is required to use his mathematical knowledge.
Note that while in each case the general problem is easy to grasp, there is not enough infor­
mation, in fact, for a modern reader to solve these mathematical problems. This is partly due 
to philological difficulties: even after two editions the text is still far from fully understood. 
The choice of this extract as the first source text is mainly meant to illustrate the social and 
cultural context of mathematics in ancient Egypt.
Papyrus Anastasi I, 13,4-18,2 
Another topic
Look, you come here and fill me with (the importance of) your office. I will let you 
know your condition when yousay: "I am the commanding scribe of the soldiers.'' 
It has been given to you to dig a lake. You come to me to ask about the rations of 
the soldiers. You say to me: “ Calculate i t ! ” I am thrown into your office. Teaching 
you to do it has fallen upon my shoulders.
I will cause you to be embarrassed, I will explain to you the command of your 
master—may he live, prosper, and be healthy. Since you are his royal scribe, you are 
sent under the royal balcony for all kinds of great monuments of Horus, the lord of 
the two lands. Look, you are the clever scribe who is at the head of the soldiers.
A ramp shall be made of (length) 730 cubits, width 55 cubits, with 120 compart­
ments, filled with reeds and beams. For height: 60 cubits at its top to 30 cubits in its 
middle, and an inclination {sqd) of 15 cubits, its base 5 cubits. Its amount of bricks 
needed shall be asked from the overseer of the troops. All the assembled scribes lack 
someone (i.e., a scribe) who knows them (i.e., the number of bricks). They trust in 
you, saying: "You are a clever scribe my friend. Decide for us quickly. Look, your 
name has come forward. One shall find someone in this place to magnify the other 
thirty. Let it not be said of you: there is something that you don't know. Answer for 
us the number (lit. its need) of bricks." Look, its measurements are before you. Each 
one of its compartments is of 30 cubits (in length) and a width of 7 cubits.
Hey Mapu, vigilant scribe, who is at the head of the soldiers, distinguished when 
you stand at the great gates, bowing beautifully under the balcony. A dispatch has 
come from the crown prince to the area of Ka to please the heart of the Horus of 
Gold, to calm the raging lion. An obelisk has been newly made, graven with the 
name of his majesty—may he live, prosper, and be healthy—of 110 cubits in the 
length of its shaft, its pedestal of 10 cubits, the circumference of its base makes 
7 cubits on all its sides, its narrowing towards the summit 1 cubit, its pyramidion 
1 cubit in height, its point 2 digits.
E gyptian M a th e m a tic s I 11
Add them up in order to make it from parts. You shall give every man to its trans­
port, those who shall be sent to the Red Mountain. Look, they are waiting for them. 
Prepare the way for the crown prince Mes-lten. Approach; decide for us the 
amount of men who will be in front of it. Do not let them repeat writing while the 
monument is in the quarry. Answer quickly, do not hesitate! Look, it is you who is 
looking for them for yourself. Get going! Look, if you hurry, I will cause your heart 
to rejoice.
I used to [...] under the top like you. Let us fight together. My heart is apt, my fin­
gers listen. They are clever, when you go astray. Go, don't cry, your helper is behind 
you. I let you say: "There is a royal scribe with Horus, the mighty bull." And you shall 
order men to make chests into which to put letters that I will have written you 
secretly. Look, it is you who shall take them for yourself. You have caused my hands 
and fingers to be trained like a bull at a feast until every feast in eternity.
You are told: "Empty the magazine that has been loaded with sand under the mon­
ument for your lord—may he live, prosper, and be healthy—which has been brought 
from the Red Mountain. It makes 30 cubits stretched upon the ground with a width of 
20 cubits, passing chambers filled with sand from the riverbank. The walls of its cham­
bers have a breadth of 4 to 4 to 4 cubits. It has a height of 50 cubits in total. [... ] You 
are commanded to find out what is before it. How many men will it take to remove it 
in 6 hours if their minds are apt? Their desire to remove it will be small if (a break at) 
noon does not come. You shall give the troops a break to receive their cakes, in order 
to establish the monument in its place. One wishes to see it beautiful.
O scribe, keen of wit, to whom nothing whatsoever is unknown. Flame in the 
darkness before the soldiers, you are the light for them. You are sent on an expedi­
tion to Phoenicia at the head of the victorious army to smite those rebels called 
Nearin. The bow-troops who are before you amount to 1900, Sherden 520, Kehek 
1600, Meshwesh <100>, Tehesi 880, sum 5000 in all, not counting their officers. A 
complimentary gift has been brought to you and placed before you: bread, cattle, 
and wine. The number of men is too great for you: the provision is too small for 
them. Sweet Kemeh bread: 300, cakes: 1800, goats of various sorts: 120, wine: 30. 
The troops are too numerous; the provisions are underrated like this what you take 
from them (i.e., the inhabitants). You receive (it); it is placed in the camp. The sol­
diers are prepared and ready. Register it quickly, the share of every man to his hand. 
The Bedouins look on in secret. O learned scribe, midday has come, the camp is hot. 
They say: 'It is time to start'. Do not make the commander angry! Long is the march 
before us. What is it, that there is no bread at all? Our night quarters are far off. What 
is it, Mapu, this beating we are receiving (lit. of us)? Nay, but you are a clever scribe. 
You cease to give (us) food when only one hour of the day has passed? The scribe 
of the ruler— may he live, prosper, and be healthy—is lacking. Were you brought to 
punish us? This is not good. If Pa-Mose hears of it, he will write to degrade you."
The extract above shows that mathematics constituted an important part in a scribe’s educa­
tion and daily life. Furthermore, it illustrates the kind of mathematics that was practiced in 
Egypt. The passages cited refer to mathematical knowledge that a scribe should have in order 
to handle his daily work: accounting of grain, land, and labor in pharaonic Egypt. There have 
been several attempts to reconstruct actual mathematical exercises from the examples referred 
to in this source. All of them have met difficulties, which are caused not only by the numerous
12 I A n n e tte Im h au sen
philological problems but also by the fact that the problems are deliberately “underdetermined.” 
These examples were not intended to be actual mathematical problems that the Egyptian 
reader (i.e., scribe) should solve, but they were meant to remind him of types of mathemati­
cal problems he encountered in his own education.
T i m e l i n e 3
Archaic Period
Dyn. 1-2 (c. 300-2686 BCE)
Old Kingdom
Dyn. 3-8 (2686-2160 BCE)
First Intermediate Period 
Dyn. 9-10 (2160-2025 BCE)
Middle Kingdom
Dyn. 11-12 (2025-1773 BCE)
Second Intermediate Period 
Dyn. 13-17 (1773-550 BCE
New Kingdom
Dyn. 18-20 (1550-1069 BCE)
Third Intermediate Period 
Dyn. 21-25 (1069-656 BCE)
Late Period
Dyn. 26-31 (664-332 BCE)
Greek/Roman Period 
(332 BCE-395 CE)
E x t a n t M a t h e m a t ic a l T e x t s ̂ S c r i p t
pMoscow (E4676)
Math. Leather Roll 
(BM10250)
Lahun Fragments 
pBerlin 6619
Cairo Wooden Boards 
pRhind (BM10057-8)
m
B:
Ostracon Senmut 153 
Ostracon Turin 57170
pCairo JE 89127-30 
PCairo JE 89137-43 
pBM 10399 
pBM10520 
pBM10794 
pCarlsberg 30
“Dates according to [Shaw 2000],
bIn this column Hieratic texts are listed in bold and italic, while Demotic texts are listed in italic.
E gyptian M a th e m a tic s I 13
Educational texts are the main source of our knowledge today about Egyptian mathemat­
ics. As already mentioned, there are very few sources available. These are listed in the table 
above. (Note that only mathematical texts, i.e., texts which teach mathematics, are included 
here, and therefore pAnastasi I is not listed.) Egyptian mathematical texts belong to two dis­
tinct groups: table texts and problem texts. Examples of both groups will be presented in this 
chapter. These are complemented by administrative texts that show mathematical practices in 
daily life.
The following paragraphs present the Egyptian number system, arithmetical techniques, 
Egyptian fraction reckoning, and metrology, in order to make the sources more easily accessible.
I.a. Inventionof writing and number systems
The earliest evidence of written texts in Egypt at the end of the fourth millennium b c e con­
sists of records of names (persons and places) as well as commodities and their quantities. 
They show the same number system as is used in later times in Egypt, a decimal system with­
out positional notation, i.e. with a new sign for every power of 10:
I D * I I ^ J4* .
1 10 100 1000 10,000 100,000 1,000,000
Naqada tablets CG 14101,14102,14103
9 ‘
n n n n 
n n n ' , 7
14101 14102
These predynastic tablets were probably attached to some commodity (there is a hole in each 
of the tablets), and represented a numeric quantity related to this commodity. The number 
written on the first tablet is 185; the sign for 100 is written once, followed by the sign for 10 
eight times, and the sign for 1 five times. The second tablet shows the number 175, and the 
third tablet 164. In addition, a necklace is drawn on the third tablet. This is interpreted as a 
tablet attached to a necklace of 164 pearls.
Parallel with the hieroglyphic script, which throughout Egyptian history was mainly used 
on stone monuments, a second, simplified script evolved, written with ink and a reed pen on 
papyrus, ostraca, leather, or wood. This cursive form of writing is known as “hieratic script.” 
The individual signs often resemble their hieroglyphic counterparts. Over time the hieratic 
script became more and more cursive, and groups of signs were combined into so-called 
ligatures.
Hieroglyphic script could be written in any direction suitable to the purpose of the inscrip­
tion, although the normal direction of writing is from right to left. Thus, the orientation of the 
individual symbols, such as the glyph for 100, varies. Compare the glyph for 100 in the table above
V X 7
n n n 11 
n o n i i
14103
9 •
n n n n
n n n n
14 I A n n e tte Im h au sen
and in the illustrated tablets. Hieratic, however, is always written from right to left. While 
hieroglyphic script is highly standardized, hieratic varies widely depending on the handwriting 
of the individual scribe. Therefore, it is customary in Egyptology to provide a hieroglyphic 
transcription of the hieratic source text.
Notation for fractions: Egyptian mathematics used unit fractions (i.e., j , etc.) almost 
exclusively; the single exception is ~. In hieratic, the number that is the denominator is writ­
ten with a dot above it to mark it as a fraction. In hieroglyphic writing the dot is replaced by 
the hieroglyph <=> (“part”). The most commonly used fractions |, j , and ^ were written by 
special signs:
hieratic
4
hieroglyphic Value
2
r 3
1
2
/ CTT7>
1
3
X X J_4
More difficult fractions like f or f were represented by sums of unit fractions written in direct 
juxtaposition, e.g., \ \ (hieroglyphic ^ x); | (hieroglyphic <s>TfT). In transcription,
fractions are rendered by the denominator with an overbar, e.g., j is written as 2. The fraction 
| is written as 3.
I.b. Arithmetic
Calculation with integers: the mathematical texts contain terms for addition, subtraction, 
multiplication, division, halving, squaring, and the extraction of a square root. Only multipli­
cation and division were performed as written calculations. Both of these were carried out 
using a variety of techniques the choice of which depended on the numerical values involved. 
The following example of the multiplication of 2000 and 5 is taken from a problem of the 
Rhind Mathematical Papyrus (remember that the hieratic original, and therefore this hiero-
glyphic transcription, are read from right to left):
Rhind Mathematical Papyrus, problem 52
\. 2 0 0 0 II ./
2 4 0 0 0 1111 n
\ 4 8 0 0 0 n nm i nn/
Total 10,000 n
The text is written in two columns. It starts with a dot in the first column and the number that 
shall be multiplied in the second column. The first line is doubled in the second line.
E gyptian M a th e m a tic s I 15
Therefore we see “2” in the first column and “4000” in the second column, the third line is 
twice the second (“4” in the first column, “8000” in the second column).
The first column is then searched for numbers that add up to the multiplicative factor 5 
(the dot in the first line counts as “1”). This can be achieved in this example by adding the first 
and third lines. These lines are marked with a checkmark (\). The result of the multiplication 
is obtained by adding the marked lines of the second column. If the multiplicative factor 
exceeds 10, the procedure is slightly modified, as can be followed in the example below from 
problem 69 of the Rhind Mathematical Papyrus. The multiplication o f 80 and 14 is performed 
as follows:
Rhind Mathematical Papyrus, problem 69
■ 80
n n n n
n n n n .
\1 0 800 999????? n /
2 160
non 
nnn 9 i
\ 4 320 n n ? 9 ? m i /
Total 1120 n n 9 J
After the initial line, we move directly to 10; then the remaining lines are carried out in the 
usual way, starting with double o f the first line.
Divisions are performed in exactly the same way, with the roles of first and second column 
switched. The following example is taken from problem 76 of the Rhind Mathematical 
Papyrus. The division that is performed is 30 -r- 2 \.
Rhind Mathematical Papyrus, problem 76
■ 22 ---- Nil
\ 10 25
I I
i n n n /
\ 2 5 11 n /
Total 12 in
Again we find two columns. This time the divisor is subsequently either doubled or multiplied 
by 10. Then the second column is searched for numbers that add up to the dividend 30. The 
respective lines are marked. The addition o f the first column of these lines leads to the result 
of the division.
Calculation with fractions: the last example of the division included a fraction (22); how­
ever, in this example it had little effect on the performance of the operation. From previous 
examples of multiplication (and division) it is obvious that doubling is an operation which has 
to be performed frequently. If fractions are involved, the fraction has to be doubled. If the
fraction is a single unit fraction with an even denominator, halving the denominator easily does 
this, e.g., double of 64 is 32 = ^ ). If> however, the denominator is odd (or a series of unit
fractions is to be doubled), the result is not as easily found. For this reason the so-called 2 -t- N 
table was created. This table lists the doubles o f odd unit fractions. Examples o f this table can 
be found in the section of table texts following this introduction. Obviously, the level of diffi­
culty in carrying out these operations usually rises considerably as soon as fractions are 
involved.
The layout of a multiplication with fractions is the same as the layout of the multiplication 
of integers. The following example— the result of which is unfortunately partly destroyed— is 
taken from problem 6 of the Rhind Mathematical Papyrus, the multiplication of 3 5 30 
with 10.
16 I A n n e tte Im h au sen
Rhind Mathematical Papyrus, problem 6
\ 2
3 5 30 hn T f r __ ,n 11 ‘T r ’ ■
13 10 30 nn
n n <=rr> " /
4 32 10 <-->
Mil
\ 8 75
M in
INI 
INI /
Total 9 loaves of bread. This is it.
9n
Note that the multiplication with ten in this case is not performed directly, but explicitly 
carried out through doubling and addition.
Divisions with a divisor greater than the dividend use a series of halvings starting either 
with | ( f , ! > . . . ) or with | (j ,\, j , ...). For instance the division 70 93 j in problem 38 of 
the Rhind Mathematical Papyrus is performed as follows:
Rhind Mathematical Papyrus, problem 58
■ 93 3
nnn 
• i i | nnn 
ill I I I nnn ■
\ 2 46 3
m nn 
<r r > i 11 n n = ^ /
\ 4 23 3 _̂_ ̂ n"xT In x /
[Total 2 4]
In more difficult numerical cases the division is first carried out as a division with remainder. 
The remainder is then handled separately.
E gyptian M a th e m a tic s I 17
l.c. Metrology
Note that the following overview is by no means a complete survey of Egyptian metrology, but 
includes only those units which areused in the sources of this chapter.
The approximate values given here are derived from the approximation that 1 cubit « 
52.5 cm, which is used in standard textbooks. It must be noted however, that this was deter­
mined as an average o f cubit rods o f so-called votive cubits, that is, cubits that have been 
placed in a tomb or temple as ritual objects. (Some other cubit rods have been unearthed 
that bear signs o f actually having been used by architects and workers.) A valid “standard 
cubit” throughout Egypt did not exist. Naturally, the same holds for area and volume 
measures.
Length measures
1 ht = 1 0 0 cubits
1 cubit = 7 palms
1 palm = 4 digits
1 digit
Area measures
1 hi-B = 1 0 st I t
1 Sti.t = (1 ht)2
1 area-cubit = 1 cubit
Volume measures
1 hlr = 16 h q lt
1 h>r = 20 hqLt = 2/3 cubic-cubit
1 hqi.t = 10 hnw
1 hnw = 32 ri
1 ri
II. Hieratic Mathematical Texts
Egyptian mathematical texts can be assigned to two groups: table texts and problem texts. 
Table texts include tables for fraction reckoning (e.g., the 2 t N table, which will be the first 
source text below, and the table found on the Mathematical Leather Roll) as well as tables for 
the conversion of measures (e.g., Rhind Mathematical Papyrus, Nos. 47, 80, and 81). Problem 
texts state a mathematical problem and then indicate its solution by means of step-by-step 
instructions. For this reason, they are also called procedure texts.
W 76.8 l3
* 96.5 l4 
* 4 . 8 1
* 0.48 1
* 15 ml
52.5 m
52.5 cm
7.5 cm 
19 mm
x 100 cubit
27562.5 m2 
2756.25 m2 
27.56 m2
3This is the New Kingdom value, from Dynasty 20 onward. In the Old Kingdom and Middle Kingdom, the value 
was 1 hlr = 10 hqi.t.
4This is the value found in the Rhind Mathematical Papyrus.
18 I A n n e tte Im h au sen
The extant hieratic source texts (in order of their publication) are
• Rhind Mathematical Papyrus (BM 10057-10058)
• Lahun Mathematical Fragments (7 fragments: UC32114, UC32118B, UC32134, U C32159-32162)
• Papyrus Berlin 6619 (2 fragments)
• Cairo Wooden Boards (CG 25367 and 25368)
• Mathematical Leather Roll (BM 10250)
• Moscow Mathematical Papyrus (E4674)
• Ostracon Senmut 153
• Ostracon Turin 57170
Most of these texts were bought on the antiquities market, and therefore we do not know their 
exact provenance. An exception is the group of mathematical fragments from Lahun, which 
were discovered by William Matthew Flinders Petrie when he excavated the Middle Kingdom 
pyramid town of Lahun.
II.a. Table texts 
UC 32159
*?7n , it ' Hi
— ^ non I i! n In 10
Reprinted by permission of Petrie Museum of Eqyptian Archaeology, University College, London.
The photograph shows a part of the so-called 2 -r N table from one of the Lahun fragments. 
The hieroglyphic transcription of the fragment on the photo is given next to it. This table 
was used to aid fraction reckoning. Remember that Egyptian fraction reckoning used only 
unit fractions and the fraction j . As multiplication consisted of repeated doubling, multi­
plication of fractions often involved the doubling of fractions. This can easily be done if the
E gyptian M a th e m a tic s I 19
denominator is even. To double a unit fraction with an even denominator, its denominator 
has to be halved, e.g., 2 x j = j .
However the doubling of a fraction with an odd denominator always consists of a series 
of two or more unit fractions, which are not self-evident. Furthermore, there are often sev­
eral possible representations; however, Egyptian mathematical texts consistently used only 
one, which can be found in the 2 -e N table. Below is the transcription of our example into 
numbers;
Column I Column II
1 2 3 3 2
2 5 3 13 15 3
3 7 4 124 28 4
4 9 6 12 18 2
5 11 6 136 66 6
6 13 8 128 52 4 104 8
7 15 K) 12 30 2
8 17 12 1312 51 3 68 4
9 19 12 1212 76 4 114
10 21 14 12 42 2
The numbers are grouped in two columns. The first column contains the divisor N (in the first 
line only it shows both dividend 2 and divisor 3). The second column shows alternatingly frac­
tions of the divisor and their value (as a series of unit fractions). For example, the second line 
starts with the divisor 5 in the first column; therefore it is 2 5 that is expressed as a series of
unit fractions. It is followed in the second column by 3, 13,15, and 3. This has to be read as 3 
of 5 is 13 and 15 of 5 is 3. Since 13 plus 3 equals 2, the series of unit fractions to represent 
2 - 5 is 3 15.
The Recto of the Rhind Mathematical Papyrus contains the 2 -r N table for N = 3 to N = 101. 
Here, the solutions are marked in red ink, rendered as bold in the transcription below. 
There have been several attempts to explain the choices of representations in the 2 e N 
table. These attempts were mostly based on modern mathematical formulas, and none 
of them gives a convincing explication of the values we find in the table. It is probable that 
the table was constructed based on experiences in handling fractions. Several “guidelines” 
for the selection of suitable fractions can be discerned. The author tried to keep the num­
ber of fractions to represent 2 -e AT small; we generally find representations composed of two 
or three fractions only. Another guiding rule seems to be the choice of fractions with a small 
denominator over a bigger denominator, and the choice of denominators that can be 
decomposed into several components.
2 0 I A n n e tte Im h au sen
Rhind Mathematical Papyrus, 2 + N Table
N 2 Hh N N 2 + N
3 3 2 53 30 13 10 318 6 795 15
5 313 15 3 55 30 136 3306
7 4124 28 4 57 3812 1142
9 612 18 2 59 36121218 236 4 5319
11 6136 66 6 61 4012402444488861010
13 8128 52 4 104 8 63 42121262
15 1012 30 2 65 39131953
17 121312 51 3 68 4 67 4012820 33555368
19 12 12 12 76 4 114 6 69 4612138 2
21 1412422 71 4012 4 40568871010
23 1213427612 73 601620219329243655
25 1513 75 3 75 50121502
27 1812542 77 44124 3084
29 24162458217462328 79 601415 237 3 316 4 790 To
31 20 12 20 124 4 1~55 5 81 54121622
33 2212662 83 601320332441554986
35 30 164236 85 51132553
37 2412 241113 296 8 87 58121742
39 2612782 89 60 1 3 10 20 356 4 534 6 890 10
41 2413 24 246 6 328 8 91 701510130330
43 42 1 42 86 2 129 3 301 7 93 62121862
45 3012902 95 601212 380 4 570 6
47 30 12 15 141 3 470 10 97 56128142867977768
49 28124 1964 99 6612 1982
51 3412 102 2 101 1011202 2 303 3 606 6
M
at
he
m
at
ic
al
 L
ea
th
er
 R
ol
l
CM I O I O I O I CM
l ^ I CO I CM I ICO
CO co co co CO _CO
4—* 4-> -t—̂ 4—' +-» 4->c
E
=3
CD I CN |0 IO I o ICO
| CM CD
CO I lo ICO I CO I CD I
C O I CD |0 I CD I00 | COCN I I CO 11— I CD
CO
c
E
o
(J
CM
c
Ej 2
o
o
ICO
CO CO 1 LO 1 T_
ICO I ICO I ID ICO I CM II CO I0 0 CO ICO 1 ^ ICO i IO ) 1 1T— 1 1
CO CO CO CO CO CO CO CO CO CO CO m CO id CO CO CO w in
4—' 4-J 4—'' 4—* 4—1 4—* 4—» +-J 4—1 4—* f - 1
1 o
1 o oo 1 M"
1 CM I O | 0
1 CD LO LO
1 1 T— 1 r— | 00
o 1 ° ICO 1 10 | 0 1 o
1 CM
1 ^
M- | 0 1 ^ ICO ICO
1 1 CO 1 LO ICO 1 CM
O 1 CM 1 CD 1 CD 1 o 1 CD 1 1 0X1
1 LO 1 1 ° ICO ICO 1 CO 1 CM I LO 1 CM 1 O) i 1 r—
IO ICO 1 ^ 1 | 0 1 ° 1 ° 1 CMT-- 1 *— 1 T— 1 T— 1 00 1 CM 1 *— 1 00
m in m CO CO I f ) CO CO
+-> 4—' 4—' 4—' 4—' 1 CO
I CNII
CM
ICO
-3- | CO cd
LO 1 CD
|C0 | cd
CM ICO
CO 1 ^ ICO1 T— 1 CM
IO 
I 05
i COI
IO
I CO
I 00
I CM
o
CD
LO too 1 co 1 CN 1 CO |C0 IO | CO
1 CM 
CD
1 ^ CO 1 05 1 CO ICO 1 CD
o 1 00 1 ID |C0 110 | coo |COCO 1 CN 1 CN 1 CO 1 1 M" 1 CT>
c
E_=S
o
o
ICO 1 ICO 1 LO ICO 1 CM II CO 1 CO
CO
ICO ICO 1
| I0 Q | ' i r ^ ' ' i cd ' 1 ^ 1 r
in cn m CO CO CO CO CO in CO CO CO m m m m
4—' 4—* 4—' 4—* 4—* 4—* 4—* 4—' 4—
I OI
I O
I O I CN
I LO
| CM 
I ^
I o0
1 CM 
I co 1D-- 
I CO
I LO 
I CN 28
 4
9 
19
6
2 2 I A n n e tte Im h au sen
The Mathematical Leather Roll is another aid for fraction reckoning. It contains 26 sums of 
unit fractions which equal a single unit fraction. The 26 sums have been noted in two 
columns, followed by another two columns with the same 26 sums. The numeric transcrip­
tion given above shows the arrangementof the sums of the source.
Apart from fraction reckoning, tables were also needed for the conversion of different 
measuring units. An example of these tables can be found in the Rhind Mathematical Papyrus, 
No. 81. Here, two systems of volume measures, hqi.t and hnw, are compared, hqi.t is the basic 
measuring unit for grain, with 1 hqi.t equaling 10 hnw. The hqi.t was used with a system of 
submultiples, which were written by distinctive signs:
2 hqi.t
4 hqi.t
X 8 hqi.t
16 hqi.t
J 32 hqi.t
i m hqht
In older literature about Egyptian mathematics these signs are often interpreted as hieratic 
versions of the hieroglyphic parts of the eye of the Egyptian god Horus. However, texts from 
the early third millennium as well as depictions in tombs of the Old Kingdom, which show the 
same signs prove that the eye of Horus was not connected to the origins of the hieratic signs.5 
1 hqi.t also equals 32 ri, the smallest unit for measuring volumes.
The table found in No. 81 of the Rhind Mathematical Papyrus is divided into three parts. 
Each part is introduced by an Egyptian particle (in the translation rendered as “now”). The 
first section of the table, arranged in two columns, lists the submultiples of the hqi.t as hnw. 
Due to the values of the submultiples, each line is half of its predecessor. The following two 
sections are both laid out in three columns. The first column gives combinations of the sub­
multiples of the hqi.t and ri. w. The second column lists the respective volume in hnw. The last 
column contains the volumes as fractions of the hqi.t, this time not written in the style of sub­
multiples but as a pure numeric fraction of the unit hqi.t.
The source text of these last two sections shows a rather large number of errors. Out of 82 
entries 11 are wrong. Some of these errors seem to be simple writing errors; some follow from 
using a faulty entry in a previous line or column to calculate the new entry. The table is given 
here with all the original (sometimes wrong) values followed by footnotes that give the cor­
rect value and— if possible— an explanation for the error. It is difficult to account for the large 
number of mistakes in this table. The Rhind Mathematical Papyrus (of which this table is a 
part) is a collection of tables and problems, mostly organized in a carefully thought out 
sequence. It was presumably the manual of a teacher.
5See [Ritter 2002] for a detailed discussion.
Rhind Mathematical Papyrus, No. 8 7 
Another reckoning of the hnw
Egyptian M a th e m a tic s I 2 3
Now
Now
Now
2 hqi.t 5
4 hqi.t 22
8 hqi.t 14
16 hqi.t 28
32 hqi.t 416
64 hqi.t 832
2 4 8 hqi.t as hnw, it is 8 2 4
24 hqi.t it is 7 2
2832 hqi.t 3 3 ri.w 6 2 166 7 it is 3 o f a hqi.t
I 00 
I CM hqi.t 64 it is 5 o f a hqi.t1
48 hqi.t 3 2 4 it is 3 o f a hqi.t8 9
432 64 hqi.t 1 3 ri.w
— 9 
3 4 8 it is 7 o f a hqi.t10 * *
4 hqi.t 22 it is 4 o f a hqi.t
816 hqi.t 4 ri.w 2 it is 5 o f a hqi.t
[§ 32] hqi.t 3 3 ri.w [ 13] it is ^6 o f J a hqi.t
8 16 hqi.t 4 ri.w it is 2 hnw it is 5 of a hqi.t
T6 32 hqi.t 2 ri.w it is 1 hnw it is 10 of a hqi.t
32 64 hqi.t 1 ri.w it is 2 hnw it is 20 of a hqi.t
64 hqi.t 3 ri.w it is 4 hnw it is 40 of a hqi.t
16 hqi.t %
ICO it is 3 hnw it is 30 of a h q i.t :
32 hqi.t 3 ri.w it is 3 hnw it is 60 of a h q i.t1
C orrect value: 63 hnw. Possible explanation for the mistake: 2 8 32 hqi.t = 6 216 hnw, therefore it is likely that
33 ri.w of the first column were forgotten when the second column was determined.
7Correct value: 28 of a hqi.t.
8Correct value: 48 of a hqi. t. _ —
9Correct value: 3 48 hnw. Possible explanation for the mistake: What I read as 4 may be a very badly written 40,
but it seems more probable to read 4.
10Correct value: 4 3248 of a hqi.t.
"Correct value: 15 of a hqi.t.
"Correct value: 30 of a hqi.t. Possible explanation for the mistake: Calculation based on wrong entry in previous 
line of this column.
2 4 I A n n e tte Im h au sen
64 hqi.t 3 ri.w it is 5 hnw]3 it is 50 of a hq i.tu
2 hqi.t it is 5 hnw it is 2 of a hqi.t
4 hqi.t it is 2 2 hnw it is 4 of a hqi.t
2 4 hqi.t it is 7 2 hnw it is 2 4 of a hqi.t
2 4 8 hqi.t it is 8 2 hnw13 14 15 it is 2 4 8 of a hqi.t
2 8 hqi.t it is 6 4 hnw it is 2 8 of a hqi.t
4 8 hqi.t it is 2 4 /zmv16 it is 4 8 of a hqi.t
2 8 32 hqi.t 3 3 ri.w it is 6 3 hnw it is 3 of a hqi.t
4 1 6 64 hqi.t 13 ri.w it is 3 3 hnw it is 3 of a hqi.t
8 hqi.t it is 1 4 hnw it is 8 of a hqi.t
II.b. Problem texts
The extant hieratic mathematical texts contain approximately 100 problems, most of which 
come from the Rhind and Moscow Mathematical Papyri. The problems can generally be 
assigned to three groups:
• pure mathematical problems teaching basic techniques
• practical problems, which contain an additional layer o f knowledge from their respective 
practical setting
• non-utilitarian problems, which are phrased with a pseudo-daily life setting without having a 
practical application (only very few examples extant)
The following sections present selected problems of all three groups. Because problems are 
often phrased elliptically, occasionally other examples from the same problem type must be 
read in order to understand the problem. This will be seen from the first two examples (Rhind 
Mathematical Papyrus, problems 26 and 27). Unfortunately, due to the scarcity of source 
material, many problem types exist only in a few examples or even'only in one.
The individual sources share a number of common features. They can generally be 
described as rhetorical, numeric, and algorithmic. “Rhetoric” refers to the texts being written 
without the use of any symbolism (like +, - , V). The complete procedure is written as a prose 
text, in which all mathematical operations are expressed verbally. “Numeric” describes the 
absence of variables (like x and y). The individual problems always use concrete numbers. 
Nevertheless, it is quite obvious that general procedures were taught through these concrete 
examples without being limited to specific numeric values. “Algorithmic” refers to the way 
mathematical knowledge was taught in Egypt— by means of procedures. The solutions to the 
problems are given as step by step instructions which lead to the numeric result of the given 
problem.
13Correct value: 6 hnw.
14Correct value: 60 of a hqi.t.
l5Correct value: 8 2 4 hnw. Possible explanation for the mistake: the author forgot to write 4.
16Correct value 3 24 hnw. Possible explanation for the mistake: the author forgot to write 3.
E gyptian M a th e m a tic s I 2 5
While the problem texts show these similarities, each source also shows some characteris­
tics which make it distinct from the others. For instance, the examples from the Rhind 
Mathematical Papyrus usually include problems, the instructions for their solution, verifica­
tion of results, and calculations related to the instructions or calculations as part of the veri­
fication. The examples from the Moscow Mathematical Papyrus only note the problem and the 
instructions for its solution. Furthermore, the two texts show slightly different ways of 
expressing these instructions. Since it is by no means self-evident to a modern reader how to 
read (and understand) these texts, the first example will be discussed in full detail. For the 
examples of practical problems, a basic knowledge of their respective backgrounds is often 
essential to understand the mathematical procedure. Therefore the commentary to those 
problems may contain an overview of their setting.
A note on language and translations
The problem texts show a high level of uniformity in grammar and wording. The individual 
parts of a problem, that is, title, announcement of its data, instructions for its solution, 
announcement and verification of the result are clearly marked through different formalisms. 
This will be mirrored in the translations given in this chapter. The individual termini for 
mathematical objects and operations were developed from daily life language. Thus the